Please help me with a Calculus problem?

Question

I have a function f(x) and I have to find the value of c (for c > 0) such that the area bounded between f(x), x = c, x = c + 1, and the x axis is maximized.

I don't want to give the function because I don't want someone giving me the answer. I want to actually know how to do it. I am in Calc II and I know this is from Calc I, but I can't remember how to solve it.

NOTE: f(x) remains positive for all of x and y.

Thanks for your time!

Answer 1

Actually, you don't have to give us the function f(x). You want to maximize

c+1

∫ f(x) dx

c

Call this g(c). Then by Leibniz' Rule,

dg/dc = f(c+1) - f(c)

dg/dc = 0 ⇒ f(c) = f(c+1)

Answer 2

The area bounded by f(x), x = c, x = c + 1, and the x-axis is just the integral from c to c + 1 of f(x) dx.  If we consider this integral as a function of c (where x is now the dummy variable for integration), then:

g(c) = Integral from c to c+1 of f(x) dx

To maximize g(c), we set its derivative equal to zero, but conveniently, the fundamental theorem of calculus tells us that the derivative (with respect to c, of course) of the right hand side of the above equation is just f(c+1) - f(c).  Therefore,

g '(c) = f(c + 1) - f(c) = 0, so

f(c) = f(c + 1).

Thus, to maximize the area under f(x) from c to c + 1, you should look for the value of c for which f(c) = f(c + 1), or equivalently you should look for zeroes of the function f(x + 1) - f(x).  This will give you suitable c values. Once you find them, just make sure to choose the c values that will maximize, and not minimize, the area.  It should be pretty simple to determine which of your values will maximize the area by just looking at the graph of the function.  Specifically, you want to choose the c values such that f(x) has a local maximum in the interval (c, c + 1).